Past exam of the physics 1 paper of the University of Oxford / 2018 / 5 / Solution

For a Lorentz transformation with $S^{\prime }$ moving at $+v$ along $x$ relative to $S$,Differentiate along the object's trajectory and divide the spatial differentials by $d{t}^{\prime }$. Using $u_x=dx/dt$ and $u_y=dy/dt$ derives the relativistic velocity-addition formula:In the rest frame of ship A, take $v=c/2$ and the Earth-frame velocity of B as $(0,-c/2)$. Here $\gamma =2/\sqrt{3}$ and the common denominator is one, soThe required relative speed is the magnitude of this two-component velocity, not either component separately:The inverse Lorentz factor check is ${\gamma }_{AB}={\gamma }_A{\gamma }_B(1-{\mathbf{u}}_A\cdot {\mathbf{u}}_B/c^2)=4/3$, which gives the same relative speed.